Optimization and Differential Topology

Course Advertisement, Fall, 1996

Math 690, Topics in Mathematics,
credits and meeting times to be arranged

Prerequisites:

Basic knowledge of standard
graduate material in algebra, linear algebra and analysis. In fact, a
good undergraduate mathematical background should be more than enough
for most of the course. Specifically, Abstract Algebra (Math 301),
Topology (Math 331), Theory of Matrices (Math 307), and Advanced
Calculus (Math 414 or 465) or consent of the instructor.

Course description:

Optimization problems appear everywhere in mathematics, science,
engineering and economics. Usually these problems are of the following
form: maximize a given real-valued function subject to a collection of
constraints. Often the constraints determine a manifold M (that is, a
surface which locally looks like cartesian n-space). The
optimization problem can then be studied by considering the appropriate
(gradient) differential equation on the manifold M. The manifold
determines many of the global properties of the corresponding flow.

Differential topology is the study of the global properties of
manifolds. One of the important tools (`Morse theory') for analyzing a
particular manifold M is the study of the real-valued functions
defined on M. We can analyze the global properties of M by
studying the global properties of optimization problems on M. We have
a powerful feedback loop between theorems about manifolds and theorems
about functions on manifolds.

We shall study the book `Differential Topology' by M. W. Hirsch
(Springer-Verlag, 1976).
`This book presents some of the basic topological ideas used in studying
differential manifolds and maps.' (from the Preface) `In today's
mathematical sciences manifolds are found in many different fields. In
algebra they occur as Lie groups; in relativity as space-time; in
economics as indifference surfaces; in mechanics as phase-spaces and
energy surfaces. Wherever dynamical processes are studied,
(hydrodynamics, population genetics, electrical circuits, etc.)
manifolds are used for the state-space, the setting for a model of the
process by a differential equation or a mapping. In most of these
examples the historical development follows the local-to-global
pattern.... The questions which differential topology tries to answer
are global; they involve the whole manifold.' (from the Introduction)

We may also study parts of the book `Surfaces' by H. B. Griffiths
(Cambridge University Press, 1976 and 1981).